 In this video, I'm going to talk about solving an equation in algebra. Basically what I'm going to be doing is I'm going to be writing the justifications for each one of these steps. So this is the beginnings of doing algebraic proofs. So the first thing we're going to do is we're going to solve this equation, okay? So I'm going to rewrite this, 1 half t plus 5 is equal to negative 7. Okay? So I'm going to rewrite this equation. Now what I'm going to do is I'm going to solve it. Now this is something that everybody can do if you're watching this video. You should be able to solve this. So what we're going to do is we're going to solve it and then we are going to basically make the justification for each one of these steps. Basically writing the reasons why we can do each one of these steps. Okay? So the first thing that I'm going to do, the first thing I'm going to do is I'm going to subtract 5 from both sides. Subtract 5 from both sides and now what that will give me is 1 half t is equal to negative 12. So I'm going to set these add to get zero so that kind of goes away. I got nothing there. Negative 7 and a negative 5 make a negative 12. All right. So there's my first step. There's my first step that I took. Okay? Now what I'm going to do is I'm going to keep solving. Now I see this 1 half here and a lot of you are thinking, okay, 1 half. 1 half is multiplying times a t so I need to divide by 1 half. Divide by 1 half. Now the tricky thing about this is we don't really know how to divide by 1 half. That's kind of tricky. Now the left side over here, that's not really that tricky at all. It's just 1 half divided by 1 half. That's going to cancel and all your left with this t. But over here, negative 12 divided by 1 half. That's a little tricky. So instead, we're going to try something a little bit different. Instead, we're going to try something just a little bit different. So give me a moment to get rid of some of these. Now what we're going to do differently is instead of, oh excuse me, dividing by this 1 half here, what we're going to do is we're actually going to multiply times 2. We're going to multiply times 2 on both sides. Now multiplying times 2 right here will cancel everything and all I have left is the t. And then whatever I do to one side, I must do to the other. So I'm going to multiply by times 2 on the right side also. Okay, so I'm going to have t equals negative 24. Okay, so that's a little bit easier to see. All right, so those are the steps that I did. Notice the steps that I have I left in red. And what we're going to do is we're going to go back and now we're going to justify those steps. What did we do during those steps? So here we go. My first justification, this first line, this was given to us. Now whatever you're doing proofs, if anything is given to you, if anything is ever given to you, that's going to be the first line of your proof. This was given to us. All right, then now from the given statement, I go down to my next step. How did I get there? How did I get from this first line to this second line? Well right here, I subtracted five from both sides. So this would be the subtraction, subtraction, property of equality. So notice the abbreviations that I'm using. Property, this is the word property, abbreviate, of equality, we just use an equal symbol for equality. Okay, so that's the first statement. That's the reason why I could do the first one. Now the next step, I go from line two here down to line three, line two to line three. What did we do here? Well, leftover is my multiply times two. So this was the multiplication property, again, abbreviating property of equality. Multiplication property of equality. Okay, so that's it, we solved the equation and we also wrote the justification for each one of the steps that we took. And that's it, that's kind of the beginnings of your algebraic proofs. All right, now moving on, what we're gonna do is I'm gonna show you another proof, but this is kind of tiptoeing into the geometry aspect of doing proofs. Okay, so solving an equation in geometry, notice we got a geometric shape over here. I know it's kind of odd to say that a line and a couple of segments, those are geometric shapes, they're simple ones, but they are geometric shapes. So we have a small segment here, which is two x, we have another small segment here, which is three x minus nine, and then we have a large segment here, which is four x minus four. So notice over here, everything's kind of spelled out for us, we have n o, which is the big segment, then we have n m, which is this small segment here on the left, and then we have m o, which is this small segment here on the right. Notice that we're adding both of these small segments to get the larger segment n o. Okay, now when I look at that, and this is a little bit different from the last proof, the last proof I said your first line has always been given. Well, this first line isn't really given to us, this is the first line and the proof, so I actually have to have a justification for this step. So what I need to do is I need to think back, what am I actually doing here? So I'm taking a small segment plus a small segment, adding those together to get a larger segment. So this is going to be segment, segment, addition, addition, postulate. Okay, now the segment addition postulate is something you probably learned a while ago, but it's still relevant to this day, we still need to know that if I take two smaller segments, add them together, I'm gonna get this larger segment, that is the segment addition postulate. So this is the justification of why I can write this. All right, now the next line, so if we're going from line one to line two, now instead of all these different letters for all the segments, now we got these expressions, two X, three X minus nine, and four X minus four. So we're taking out the letters and we're plugging in these expressions. So that kind of gives you a hint there on what this is going to be, this is going to be the substitution property. Substitution, if I can spell correctly, property, abbreviate that, of equality, substitution property of equality, taking out the segment letters and then plugging in the expressions. Okay, now after that, notice that we go from line two to line three down here, now this four X minus four over here doesn't change, but if you look over here, we have now we have five X minus nine, the nine didn't change, but these changed right here. Now this isn't necessarily a property, this is what we call simplifying. Now there's two ways you can write this, this is either called simplifying or I actually like to call this combining like terms, combining like terms, combining like terms. That's what I usually like to call this, that's what we did, we took the two X and the three X, added them together to get five X. So that's again the justification for the step, that's what we did. All right, so now line three to line four. Now notice, what did we do here? Now one way to look at what we did here is try to find out what's absent, try to find out what's not there. So as I look here, there's a definite gap right here, this four X is no longer there. So how did we get rid of it? Well if you look to the other side, five X became X. So it looks like we took the four X and subtracted it over to the other side. Okay, we took the four X and we subtracted it over to the other side. Let's look at what we did. It's gonna disappear here over on the left side but then over on the right side, we're gonna get X. So notice the step that we did there, we subtracted from both sides. That's going to give us the justification of the subtraction, subtraction, property of equality, subtraction, property of equality. Okay, the steps that you take give you a really big hint, really big hint on what your property is going to be. Okay, so now again, now we're going from these, one, two, three, four, we're on the fourth line to the fifth line, four to five, four to five. On the fourth line, again, let's see what's absent. It's like that number changed, the X is still there, but then the negative nine is absent. So it looks like what we did here is we actually added nine to this side and added nine to that side is negative four and a positive nine create a five. So that makes sense, that makes sense of what we did. Okay, and then this right here is gonna make zero so all I have left is the X. So it looks like we added to both sides, this is going to be the addition, the addition, property of equality, the addition, property of equality was that step right there. Okay, last but not least, now this one's a little odd, this one's a little odd looking. Looks like we just kind of flip things around, X went on the left side, five went on the right side. Kind of an interesting property where we kind of switch things around as the same thing on the left as the same thing on the right. This would be the symmetric property. Symmetric, symmetric property of equality, okay? So this one would be the symmetric property of equality. That says I can flip around the left and the right side of my equality and everything's gonna stay the same. When I was in high school, I was one of those kids who always wanted to write the variable first. I was one of those sticklers, I always had to have the variable first. It just, it made a whole lot more sense to have the variable first as opposed to having it second. I really didn't like to have that second. So I was able to move that around because of the symmetric property of equality. That allows me to be able to do that. Okay, just flip-flopping the sides of an equation. Alrighty, that's a couple of examples. Solving an equation using algebra, okay, like an algebra proof, very simple algebra proof, and then also solving an equation in geometry using a tad bit of geometry to help us with this proof. Now notice the tad bit of geometry is right here, the segment addition postulate that we use. So the figure is a couple of segments up here, three segments in fact, small one here, small one there, and then the big segments when we add them together. Substitution property of equality, we use that for one of our steps. Now simplifying and combining like terms, that one's a little bit different. That's not necessarily a property that you'll see all the time, but it is one of the steps that we are allowed to do. I like to say combining like terms, because that's more of a mathematical steps. It's more of a mathy word to use. I like to use that a lot better. Okay, then we had the subtraction property of equality. We subtracted from both sides. We had the addition property of equality when we added something to both sides, and then we had the symmetric property where we flip-flopped these last two statements. The five went to the right and the x went to the left. We kind of flip-flopped everything. All right, so those are a couple of examples of your entry-level algebraic proofs.